Computed tomography (CT) systems and methods are widely used, particularly for medical imaging and diagnosis. CT systems generally create images of one or more sectional slices through a subject's body. A radiation source, such as an X-ray source, irradiates the body from one side. At least one detector on the opposite side of the body receives radiation transmitted through the body. The attenuation of the radiation that has passed through the body is measured by processing electrical signals received from the detector.
A CT sinogram indicates attenuation through the body as a function of position along a detector array and as a function of the projection angle between the X-ray source and the detector array for various projection measurements. In a sinogram, the spatial dimensions refer to the position along the array of X-ray detectors. The time/angle dimension refers to the projection angle of X-rays, which changes as a function of time during a CT scan. The attenuation resulting from a portion of the imaged object (e.g., a vertebra) will trace out a sine wave around the vertical axis. Those portions farther from the axis of rotation correspond to sine waves with larger amplitudes, and the phase of the sine waves correspond to the angular positions of objects around the rotation axis. Performing an inverse Radon transform—or any other image reconstruction method—reconstructs an image from the projection data in the sinogram. Two families of reconstructions methods commonly used in clinical applications are filtered-back projection (FBP) and iterative reconstruction (IR).
There has arisen a push to reduce the radiation dose of clinical CT scans to become as low as reasonably achievable. Thus, iterative image reconstruction has been playing a more significant role in CT imaging. Iterative image reconstruction algorithms, as compared with traditional analytical algorithms, are promising in reducing the radiation dose while improving the CT image quality.
In X-ray computed tomography (CT), iterative reconstruction can be used to generate images. While various IR methods exist, one common IR method is optimizing the expression
      argmin    x    ⁢      {                                                  x            -            y                                    W        2            +              β        ⁢                                  ⁢                  U          ⁡                      (            x            )                                }  to obtain the argument x that minimize the expression. For example, in X-ray CT A is the system matrix that represents X-ray trajectories (i.e., line integrals) along various rays from a source through an object OBJ to an X-ray detector (e.g., the X-ray transform corresponding to projections through the three-dimensional object OBJ onto a two-dimensional projection image y), y represents projection images taken at a series of projection angles and corresponding to the log-transform of the measured X-ray intensity at the X-ray detector, and x represents the reconstructed image of the X-ray attenuation of the object OBJ. The notation ∥g∥W2 signifies a weighted inner product of the form gTWg, wherein W is the weight matrix. For example, the weight matrix W can weigh the pixel values according to their noise statistics (e.g., the signal-to-noise ratio), in which case the weight matrix W is diagonal when the noise of each pixel is statistically independent. The data-fidelity term ∥Ax−y∥W2 is minimized when the forward projection A of the reconstructed image x provides a good approximation to all measured projection images y. In the above expression, U(x) is a regularization term, and β is a regularization parameter that determines the relative contributions of the data-fidelity term and the regularization term.
IR methods augmented with regularization can have several advantages over other reconstruction methods such as filtered back-projection. For example, IR methods augmented with regularization can produce high-quality reconstructions for few-view projection data and in the presence of significant noise. For few-view, limited-angle, and noisy projection scenarios, the application of regularization operators between reconstruction iterations seeks to tune the final and/or intermediate results to some a priori model. For example, minimizing the “total variation” (TV) in conjunction with projection on convex sets (POCS) is also a very popular regularization scheme. The TV-minimization algorithm assumes that the image is predominantly uniform over large regions with sharp transitions at the boundaries of the uniform regions, which is generally true for an image of a discrete number of organs, each with an approximately constant X-ray absorption coefficient (e.g., bone having a first absorption coefficient, the lungs having second coefficient, and the heart having a third coefficient). When the a priori model corresponds well to the image object OBJ, these regularized IR algorithms can produce good image quality even though the reconstruction problem is significantly underdetermined (e.g., few-view scenarios), missing projection angles, or noisy.
If image reconstruction is performed using IR without weighting (e.g., where the weighting matrix W is replaced with an identity matrix), then streak artifacts appear in the reconstructed image. While the streak artifacts can be largely mitigated in IR by using weights that depend, at least in part, on the statistical properties of the data, the weights tend to favor tangential rays passing through the periphery of the subject because these rays are attenuated less, resulting in larger signal-to-noise ratios (SNR) for the tangential rays, which results in a larger weights and a disproportionately large contribution to the reconstructed image due to the tangential rays.